In this work, we consider linear and nonlinear Volterra integral equations of the second kind. Here, by converting integral equation of the first kind to a linear equation of the second kind and the ordinary differential equation to integral equation we are going to solve the equation easily. The Adomian decomposition method or shortly (ADM) is used to find a solution to these equations. The Adomian decomposition method converts the Volterra integral equations into determination of computable components. The existence and uniqueness of solutions of linear (or nonlinear) Volterra integral equations of the second kind are expressed by theorems. If an exact solution exists for the problem, then the obtained series convergence very rabidly to that solution. A nonlinear term F(u) in nonlinear volterra integral equations is Lipschitz continuous and has polynomial representation. Finally, the sufficient condition that guarantees a unique solution of Volterra (linear and nonlinear) integral equations with the choice of the initial data is obtained, and the solution is found in series form. Theoretical considerations are being discussed. To illustrate the ability and simplicity of the method. A few examples including linear and nonlinear are provided to show validity and applicability of this approach. The results are taken from the works mentioned in the reference.
Published in | American Journal of Applied Mathematics (Volume 6, Issue 4) |
DOI | 10.11648/j.ajam.20180604.12 |
Page(s) | 142-148 |
Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
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Copyright © The Author(s), 2018. Published by Science Publishing Group |
Linear Integral Equations, Fredholm Integral Equations, Regularization Method, Direct Computation Method, Two - Dimensional Integral Equations
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[8] | H. Jafari, E. Tayyebi1, S. Sadeghi, C. M. Khalique, A new modification of the Adomian decomposition method for nonlinear integral equations, Int. J. Adv. Appl. Math, and Mech. 1(4) (2014) 33 – 39. |
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APA Style
Ali Elhrary Abaoub, Abejela Salem Shkheam, Suad Mawloud Zali. (2018). The Adomian Decomposition Method of Volterra Integral Equation of Second Kind. American Journal of Applied Mathematics, 6(4), 142-148. https://doi.org/10.11648/j.ajam.20180604.12
ACS Style
Ali Elhrary Abaoub; Abejela Salem Shkheam; Suad Mawloud Zali. The Adomian Decomposition Method of Volterra Integral Equation of Second Kind. Am. J. Appl. Math. 2018, 6(4), 142-148. doi: 10.11648/j.ajam.20180604.12
AMA Style
Ali Elhrary Abaoub, Abejela Salem Shkheam, Suad Mawloud Zali. The Adomian Decomposition Method of Volterra Integral Equation of Second Kind. Am J Appl Math. 2018;6(4):142-148. doi: 10.11648/j.ajam.20180604.12
@article{10.11648/j.ajam.20180604.12, author = {Ali Elhrary Abaoub and Abejela Salem Shkheam and Suad Mawloud Zali}, title = {The Adomian Decomposition Method of Volterra Integral Equation of Second Kind}, journal = {American Journal of Applied Mathematics}, volume = {6}, number = {4}, pages = {142-148}, doi = {10.11648/j.ajam.20180604.12}, url = {https://doi.org/10.11648/j.ajam.20180604.12}, eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20180604.12}, abstract = {In this work, we consider linear and nonlinear Volterra integral equations of the second kind. Here, by converting integral equation of the first kind to a linear equation of the second kind and the ordinary differential equation to integral equation we are going to solve the equation easily. The Adomian decomposition method or shortly (ADM) is used to find a solution to these equations. The Adomian decomposition method converts the Volterra integral equations into determination of computable components. The existence and uniqueness of solutions of linear (or nonlinear) Volterra integral equations of the second kind are expressed by theorems. If an exact solution exists for the problem, then the obtained series convergence very rabidly to that solution. A nonlinear term F(u) in nonlinear volterra integral equations is Lipschitz continuous and has polynomial representation. Finally, the sufficient condition that guarantees a unique solution of Volterra (linear and nonlinear) integral equations with the choice of the initial data is obtained, and the solution is found in series form. Theoretical considerations are being discussed. To illustrate the ability and simplicity of the method. A few examples including linear and nonlinear are provided to show validity and applicability of this approach. The results are taken from the works mentioned in the reference.}, year = {2018} }
TY - JOUR T1 - The Adomian Decomposition Method of Volterra Integral Equation of Second Kind AU - Ali Elhrary Abaoub AU - Abejela Salem Shkheam AU - Suad Mawloud Zali Y1 - 2018/11/01 PY - 2018 N1 - https://doi.org/10.11648/j.ajam.20180604.12 DO - 10.11648/j.ajam.20180604.12 T2 - American Journal of Applied Mathematics JF - American Journal of Applied Mathematics JO - American Journal of Applied Mathematics SP - 142 EP - 148 PB - Science Publishing Group SN - 2330-006X UR - https://doi.org/10.11648/j.ajam.20180604.12 AB - In this work, we consider linear and nonlinear Volterra integral equations of the second kind. Here, by converting integral equation of the first kind to a linear equation of the second kind and the ordinary differential equation to integral equation we are going to solve the equation easily. The Adomian decomposition method or shortly (ADM) is used to find a solution to these equations. The Adomian decomposition method converts the Volterra integral equations into determination of computable components. The existence and uniqueness of solutions of linear (or nonlinear) Volterra integral equations of the second kind are expressed by theorems. If an exact solution exists for the problem, then the obtained series convergence very rabidly to that solution. A nonlinear term F(u) in nonlinear volterra integral equations is Lipschitz continuous and has polynomial representation. Finally, the sufficient condition that guarantees a unique solution of Volterra (linear and nonlinear) integral equations with the choice of the initial data is obtained, and the solution is found in series form. Theoretical considerations are being discussed. To illustrate the ability and simplicity of the method. A few examples including linear and nonlinear are provided to show validity and applicability of this approach. The results are taken from the works mentioned in the reference. VL - 6 IS - 4 ER -